Local semiconvexity of Kantorovich potentials on non-compact manifolds

Alessio Figalli; Nicola Gigli

ESAIM: Control, Optimisation and Calculus of Variations (2011)

  • Volume: 17, Issue: 3, page 648-653
  • ISSN: 1292-8119

Abstract

top
We prove that any Kantorovich potential for the cost function c = d2/2 on a Riemannian manifold (M, g) is locally semiconvex in the “region of interest”, without any compactness assumption on M, nor any assumption on its curvature. Such a region of interest is of full μ-measure as soon as the starting measure μ does not charge n – 1-dimensional rectifiable sets.

How to cite

top

Figalli, Alessio, and Gigli, Nicola. "Local semiconvexity of Kantorovich potentials on non-compact manifolds." ESAIM: Control, Optimisation and Calculus of Variations 17.3 (2011): 648-653. <http://eudml.org/doc/272798>.

@article{Figalli2011,
abstract = {We prove that any Kantorovich potential for the cost function c = d2/2 on a Riemannian manifold (M, g) is locally semiconvex in the “region of interest”, without any compactness assumption on M, nor any assumption on its curvature. Such a region of interest is of full μ-measure as soon as the starting measure μ does not charge n – 1-dimensional rectifiable sets.},
author = {Figalli, Alessio, Gigli, Nicola},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Kantorovich potential; optimal transport; regularity},
language = {eng},
number = {3},
pages = {648-653},
publisher = {EDP-Sciences},
title = {Local semiconvexity of Kantorovich potentials on non-compact manifolds},
url = {http://eudml.org/doc/272798},
volume = {17},
year = {2011},
}

TY - JOUR
AU - Figalli, Alessio
AU - Gigli, Nicola
TI - Local semiconvexity of Kantorovich potentials on non-compact manifolds
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2011
PB - EDP-Sciences
VL - 17
IS - 3
SP - 648
EP - 653
AB - We prove that any Kantorovich potential for the cost function c = d2/2 on a Riemannian manifold (M, g) is locally semiconvex in the “region of interest”, without any compactness assumption on M, nor any assumption on its curvature. Such a region of interest is of full μ-measure as soon as the starting measure μ does not charge n – 1-dimensional rectifiable sets.
LA - eng
KW - Kantorovich potential; optimal transport; regularity
UR - http://eudml.org/doc/272798
ER -

References

top
  1. [1] L. Ambrosio, N. Fusco and D. Pallara, Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York (2000). Zbl0957.49001MR1857292
  2. [2] L. Ambrosio, N. Gigli and G. Savaré, Gradient flows in metric spaces and in spaces of probability measures, Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel (2005). Zbl1090.35002MR2129498
  3. [3] D. Cordero-Erasquin, R.J. McCann and M. Schmuckenschlager, A Riemannian interpolation inequality à la Borell, Brascamp and Lieb. Invent. Math.146 (2001) 219–257. Zbl1026.58018MR1865396
  4. [4] A. Fathi and A. Figalli, Optimal transportation on non-compact manifolds. Israel J. Math. (to appear). Zbl1198.49044MR2607536
  5. [5] A. Figalli, Existence, uniqueness, and regularity of optimal transport maps. SIAM J. Math. Anal.39 (2007) 126–137. Zbl1132.28322MR2318378
  6. [6] W. Gangbo and R.J. McCann, The geometry of optimal transportation. Acta Math.177 (1996) 113–161. Zbl0887.49017MR1440931
  7. [7] N. Gigli, Second order analysis on ( 𝒫 2 ( M ) , W 2 ) . Memoirs of the AMS (to appear), available at http://cvgmt.sns.it/cgi/get.cgi/papers/gig09/. Zbl1253.58008
  8. [8] R.J. McCann, Polar factorization of maps on Riemannian manifolds. Geom. Funct. Anal.11 (2001) 589–608. Zbl1011.58009MR1844080
  9. [9] C. Villani, Optimal transport, old and new, Grundlehren des mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 338. Springer-Verlag, Berlin-New York (2009). Zbl1156.53003MR2459454

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.